Let $e_n(x_1,x_2,x_3,\dots)$ denote the $n$-th elementary symmetric function in the infinite variables $x_1,x_2,x_3,\dots$.

Let $u$ and $v$ be the roots of $z^2-6z+1=0$.

Question.Let $x_j=\frac1{j^8}$. The following seems to be true, but can one prove or disprove? $$e_n(x_1,x_2,x_3,\dots)=\frac{4^{3n+1}\pi^{8n}(u^{2n+1}+v^{2n+1})}{(8n+4)!}.$$

For example, $e_0=1$ and $e_1=\sum_{j\geq1}\frac1{j^8}=\zeta(8)=\frac{\pi^8}{9450}$.